Asymmetric Localization by Second-Harmonic Generation

We introduce a nonlinear photonic system that enables asymmetric localization and unidirectional transfer of an electromagnetic wave through the second-harmonic generation process. Our proposed scattering setup consists of a noncentrosymmetric nonlinear slab with nonlinear susceptibility χ(2) placed to the left side of a one-dimensional periodic linear photonic crystal with an embedded defect. We engineered the linear lattice to allow the localization of a selected frequency 2ω⋆ while frequency ω⋆ is in the gap. Thus in our proposed scattering setup, a left-incident coherent transverse electric wave with frequency ω⋆ partially converts to frequency 2ω⋆ and becomes localized at the defect layer while the unconverted remaining field with frequency ω⋆ exponentially decays throughout the lattice and gets reflected. For a right-incident wave with frequency ω⋆ there will not be any frequency conversion and the incident wave gets fully reflected. Our proposed structure will find application in designing optical components such as optical sensors, switches, transistors, and logic elements

Asymmetric Localization by Second-Harmonic Generation

We introduce a nonlinear photonic system that enables asymmetric localization and unidirectional transfer of an electromagnetic wave through the second-harmonic generation process. Our proposed scattering setup consists of a noncentrosymmetric nonlinear slab with nonlinear susceptibility χ(2) placed to the left side of a one-dimensional periodic linear photonic crystal with an embedded defect. We engineered the linear lattice to allow the localization of a selected frequency 2ω⋆ while frequency ω⋆ is in the gap. Thus in our proposed scattering setup, a left-incident coherent transverse electric wave with frequency ω⋆ partially converts to frequency 2ω⋆ and becomes localized at the defect layer while the unconverted remaining field with frequency ω⋆ exponentially decays throughout the lattice and gets reflected. For a right-incident wave with frequency ω⋆ there will not be any frequency conversion and the incident wave gets fully reflected. Our proposed structure will find application in designing optical components such as optical sensors, switches, transistors, and logic elements

Scattering Theory and PT\mathcal{P}\mathcal{T}-Symmetry

We outline a global approach to scattering theory in one dimension that allows for the description of a large class of scattering systems and their $\mathcal{P}$-, $\mathcal{T}$-, and $\mathcal{P}\mathcal{T}$-symmetries. In particular, we review various relevant concepts such as Jost solutions, transfer and scattering matrices, reciprocity principle, unidirectional reflection and invisibility, and spectral singularities. We discuss in some detail the mathematical conditions that imply or forbid reciprocal transmission, reciprocal reflection, and the presence of spectral singularities and their time-reversal. We also derive generalized unitarity relations for time-reversal-invariant and $\mathcal{P}\mathcal{T}$-symmetric scattering systems, and explore the consequences of breaking them. The results reported here apply to the scattering systems defined by a real or complex local potential as well as those determined by energy-dependent potentials, nonlocal potentials, and general point interactions.Comment: Slightly expanded revised version, 38 page